A direct‐forcing pressure‐based lattice Boltzmann method for solving fluid–particle interaction problems

A direct‐forcing immersed boundary‐lattice Boltzmann method (IB–LBM) is developed to simulate fluid–particle interaction problems. This method uses the pressure‐based LBM to solve the incompressible flow field and the immersed boundary method to handle the fluid–particle interactions. The pressure‐based LBM uses the pressure distribution functions instead of the density distribution functions as the independent dynamic variables. The main idea is to explicitly eliminate the compressible effect due to the density fluctuation. In the IB method, a direct‐forcing method is introduced to capture the particle motion. It directly computes an IB force density at each lattice grid from the differences between the pressure distribution functions obtained by the LBM and the equilibrium pressure distribution functions computed from the particle velocity. By applying this direct‐forcing method, the IB–LBM becomes a purely LBM version. Also, by applying the Gauss theorem, the formulas for computing the force and the torque acting on the particle from the flows are derived from the volume integrals over the particle volume instead of from the surface integrals over the particle surface. The order of accuracy of the IB–LBM is demonstrated on the errors of velocity field, wall stress, and gradients of velocity and pressure. As a demonstration of the efficiency and capabilities of the new method, sedimentation of a large number of spherical particles in an enclosure is simulated. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

A variant of immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this paper to simulate incompressible viscous flows around moving objects. As compared with the conventional IB‐LBM where the force density is computed explicitly by Hook's law or the direct forcing method and the non‐slip condition is only approximately satisfied, in the present work, the force density term is considered as the velocity correction which is determined by enforcing the non‐slip condition at the boundary. The lift and drag forces on the moving object can be easily calculated via the velocity correction on the boundary points. The capability of the present method for moving objects is well demonstrated through its application to simulate flows around a moving circular cylinder, a rotationally oscillating cylinder, and an elliptic flapping wing. Furthermore, the simulation of flows around a flapping flexible airfoil is carried out to exhibit the ability of the present method for implementing the elastic boundary condition. It was found that under certain conditions, the flapping flexible airfoil can generate larger propulsive force than the flapping rigid airfoil. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid immersed‐boundary and multi‐block lattice Boltzmann method for simulating fluid and moving‐boundaries interactions

A numerical method is developed for modelling the interactions between incompressible viscous fluid and moving boundaries. The principle of this method is introducing the immersed‐boundary concept in the framework of the lattice Boltzmann method, and improving the accuracy and efficiency of the simulation by refining the mesh near moving boundaries. Besides elastic boundary with a constitutive law, the method can also efficiently simulate solid moving‐boundary interacting with fluid by employing the direct forcing technique. The method is validated by the simulations of flow past a circular cylinder, two cylinders moving with respect to each other and flow around a hovering wing. The versatility of the method is demonstrated by the numerical studies including elastic filament flapping in the wake of a cylinder and fish‐like bodies swimming in quiescent fluid. Copyright © 2006 John Wiley & Sons, Ltd.     

An improved immersed boundary‐lattice Boltzmann method based on force correction technique

In this paper, an improved immersed boundary‐lattice Boltzmann method based on the force correction technique is presented for fluid‐structure interaction problems including the moving boundary interfaces. By introducing a force correction coefficient, the non‐slip boundary conditions are much better enforced compared with the conventional immersed boundary‐lattice Boltzmann methods. In addition, the implicit and iterative calculations are avoided; thus, the computational cost is reduced dramatically. Several numerical experiments are carried out to test the efficiency of the method. It is found that the method has the second‐order accuracy, and the non‐slip boundary conditions are enforced indeed. The numerical results also show that the present method is a suitable tool for fluid‐structure interaction problems involving complex moving boundaries.     

Simulation of three‐dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method

An improved immersed boundary–lattice Boltzmann method (IB–LBM) developed recently [28] was applied in this work to simulate three‐dimensional (3D) flows over moving objects. By enforcing the non‐slip boundary condition, the method could avoid any flow penetration to the wall. In the developed IB–LBM solver, the flow field is obtained on the non‐uniform mesh by the efficient LBM that is based on the second‐order one‐dimensional interpolation. As a consequence, its coefficients could be computed simply. By simulating flows over a stationary sphere and torus [28] accurately and efficiently, the proposed IB–LBM showed its ability to handle 3D flow problems with curved boundaries. In this paper, we further applied this method to simulate 3D flows around moving boundaries. As a first example, the flow over a rotating sphere was simulated. The obtained results agreed very well with the previous data in the literature. Then, simulation of flow over a rotating torus was conducted. The capability of the improved IB–LBM for solving 3D flows over moving objects with complex geometries was demonstrated via the simulations of fish swimming and dragonfly flight. The numerical results displayed quantitative and qualitative agreement with the date in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

An improved Rhie–Chow interpolation scheme for the smoothed‐interface immersed boundary method

Rhie–Chow interpolation is a commonly used method in CFD calculations on a co‐located mesh in order to suppress non‐physical pressure oscillations arising from chequerboard effects. A fully parallelized smoothed‐interface immersed boundary method on a co‐located grid is described in this paper. We discuss the necessity of modifications to the original Rhie–Chow interpolation in order to deal with a locally refined mesh. Numerical simulation with the modified scheme of Choi shows that numerical dissipation due to Rhie–Chow interpolation introduces significant errors at the immersed boundary. To address this issue, we develop an improved Rhie–Chow interpolation scheme that is shown to increase the accuracy in resolving the flow near the immersed boundary. We compare our improved scheme with the modified scheme of Choi by parallel simulations of benchmark flows: (i) flow past a stationary cylinder; (ii) flow past an oscillating cylinder; and (iii) flow past a stationary elliptical cylinder, where Reynolds numbers are tested in the range 10–200. Our improved scheme is significantly more accurate and compares favourably with a staggered grid algorithm. We also develop a scheme to compute the boundary force for the direct‐forcing immersed boundary method efficiently. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparative study of lattice Boltzmann methods using bounce‐back schemes and immersed boundary ones for flow acoustic problems

In order to find applicable treatments of moving boundary conditions based on the lattice Boltzmann method in flow acoustic problems, three bounce‐back (BB) methods and four kinds of immersed boundary (IB) methods are compared. We focused on fluid–solid boundary conditions for flow acoustic problems especially the simulations of sound waves from moving boundaries. BB methods include link bounce‐back, interpolation bounce‐back and unified interpolation bounce‐back methods. Five IB methods are explicit and implicit direct‐forcing (Explicit‐IB and Implicit‐IB), two kinds of partially saturated computational methods and ghost fluid method. In order to reduce the spurious pressure generated by the fresh grid node changing from solid domain to fluid domain for BB methods and sharp IB methods, we proposed two new kinds of treatments and compared them with two existing ones. Simulations of the benchmark problems prove that the local evolutionary iteration (LI) is the best one in treatments of the fresh nodes. In addition, for standing boundary problems, although BB methods have a little higher accuracy, all the methods have similar accuracy. However, for moving boundary problems, IB methods are more appropriate than BB methods, because IB methods' smooth interpolation of pressure eld produces less disturbing spurious pressure waves. With improved treatments of fresh nodes, BB methods are also acceptable for moving boundary acoustic problems. In comparative tests in respective type, unified interpolation bounce‐back with LI, Implicit‐IB, and ghost fluid with LI are the best choices. Copyright © 2013 John Wiley & Sons, Ltd.     

An immersed boundary-lattice Boltzmann flux solver and its applications to fluid–structure interaction problems

An immersed boundary-lattice Boltzmann flux solver (IB–LBFS) for the simulation of two-dimensional fluid–structure interaction (FSI) problems is presented in this paper. The IB–LBFS applies the fractional-step method to split the overall solution process into the predictor step and the corrector step. In the predictor step, the intermediate flow field is predicted by applying the LBFS (lattice Boltzmann flux solver) without considering the presence of immersed object. The LBFS applies the finite volume method to solve N–S (Navier–Stokes) equations for the flow variables at cell centers. At each cell interface, the LBFS evaluates its viscous and inviscid fluxes simultaneously through local reconstruction of the LBE (lattice Boltzmann equation) solutions. In the corrector step, the intermediate flow field is corrected by the implicit boundary condition-enforced immersed boundary method (IBM) so that the no-slip boundary conditions can be accurately satisfied. The IB–LBFS effectively combines the advantages of the LBFS in solving the flow field and the flexibility of the IBM in dealing with boundary conditions. Consequently, the IB–LBFS presents a much simpler and more effective approach for simulating complex FSI problems on non-uniform grids. Several test cases, including flows past one and two cylinders with prescribed motions, are firstly simulated to examine the accuracy of present solver. After that, two strongly coupled fluid–structure interaction problems, i.e., particle sedimentations and vortex-induced vibrations of a circular cylinder are investigated. Good agreements between the present results and those in literature verify the capability and flexibility of IB–LBFS for simulating FSI problems.     

Simulating 3D deformable particle suspensions using lattice Boltzmann method with discrete external boundary force

A method for direct numerical analysis of three‐dimensional deformable particles suspended in fluid is presented. The flow is computed on a fixed regular ‘lattice’ using the lattice Boltzmann method (LBM), where each solid particle is mapped onto a Lagrangian frame moving continuously through the domain. Instead of the bounce‐back method, an external boundary force (EBF) is used to impose the no‐slip boundary condition at the fluid–solid interface for stationary or moving boundaries. The EBF is added directly to the lattice Boltzmann equation. The motion and orientation of the particles are obtained from Newtonian dynamics equations. The advantage of this approach is outlined in comparison with the standard and higher‐order interpolated bounce‐back methods as well as the LBM immersed‐boundary and the volume‐of‐fluid methods. Although the EBF method is general, in this application, it is used in conjunction with the lattice–spring model for deformable particles. The methodology is validated by comparing with experimental and theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

A filter‐based,mass‐conserving lattice Boltzmann method for immiscible multiphase flows

Some issues of He–Chen–Zhang lattice Boltzmann equation (LBE) method (referred as HCZ model) (J. Comput. Physics 1999; 152 :642–663) for immiscible multiphase flows with large density ratio are assessed in this paper. An extended HCZ model with a filter technique and mass correction procedure is proposed based on HCZ's LBE multiphase model. The original HCZ model is capable of maintaining a thin interface but is prone to generating unphysical oscillations in surface tension and index function at moderate values of density ratio. With a filtering technique, the monotonic variation of the index function across the interface is maintained with larger density ratio. Kim's surface tension formulation for diffuse–interface method (J. Comput. Physics 2005; 204 :784–804) is then used to remove unphysical oscillation in the surface tension. Furthermore, as the density ratio increases, the effect of velocity divergence term neglected in the original HCZ model causes significant unphysical mass sources near the interface. By keeping the velocity divergence term, the unphysical mass sources near the interface can be removed with large density ratio. The long‐time accumulation of the modeling and/or numerical errors in the HCZ model also results in the error of mass conservation of each dispersed phase. A mass correction procedure is devised to improve the performance of the method in this regard. For flows over a stationary and a rising bubble, and capillary waves with density ratio up to 100, the present approach yields solutions with interface thickness of about five to six lattices and no long‐time diffusion, significantly advancing the performance of the LBE method for multiphase flow simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

A coupled immersed boundary‐lattice Boltzmann method with smoothed point interpolation method for fluid‐structure interaction problems

The immersed boundary‐lattice Boltzmann method has been verified to be an effective tool for fluid‐structure interaction simulation associated with thin and flexible bodies. The newly developed smoothed point interpolation method (S‐PIM) can handle the largely deformable solids owing to its softened model stiffness and insensitivity to mesh distortion. In this work, a novel coupled method has been proposed by combining the immersed boundary‐lattice Boltzmann method with the S‐PIM for fluid‐structure interaction problems with large‐displacement solids. The proposed method preserves the simplicity of the lattice Boltzmann method for fluid solvers, utilizes the S‐PIM to establish the realistic constitutive laws for nonlinear solids, and avoids mesh regeneration based on the frame of the immersed boundary method. Both two‐ and three‐dimensional numerical examples have been carried out to validate the accuracy, convergence, and stability of the proposed method in consideration of comparative results with referenced solutions.     

An evaluation of force terms in the lattice Boltzmann models in simulating shallow water flows over complex topography

Proper approximation of the force terms, especially the bed slope term, is of crucial importance to simulating shallow water flows in lattice Boltzmann (LB) models. However, there is little discussion on the schemes of adding force terms to LB models for shallow water equations (SWEs). In this study, we evaluate the performance of forcing schemes coupled with different LB models (LABSWE and MLBSWE) in simulating shallow water flows over complex topography and try to find out their intrinsic characteristics and applicability. Three cases are adopted for evaluation, including a stationary case, a one-dimensional tidal wave flow over an irregular bed, and a steady flow over a two-dimensional seamount. The simulating results are compared with analytical solutions or the results produced by the finite difference method. For LABSWE, all the forcing schemes, except for the weighting factor method, fail to produce accurate solutions for the test cases; this is probably due to the mismatch between the bed slope term in source terms and the quadratic depth term of the equilibrium distribution functions in these forcing schemes. For MLBSWE, all the forcing schemes are capable of simulating flows over the complex topography accurately; furthermore, those schemes taking into account the collision effect τ to eliminate the momentum induced by forces provide more accurate solutions with quicker convergence as the lattice size decreases. In this view, MLBSWE can bring more flexibility in treating the force terms and thus can be a better tool to simulate shallow water flows over complex topography in practical application.     

A hybrid phase field multiple relaxation time lattice Boltzmann method for the incompressible multiphase flow with large density contrast

A hybrid phase field multiple relaxation time lattice Boltzmann method (LBM) is presented in this paper for simulation of multiphase flows with large density contrast. In the present method, the flow field is solved by a lattice Boltzmann equation. Concurrently, the interface of two fluids is captured by solving the macroscopic Cahn‐Hilliard equation using the upwind scheme. To be specific, for simulation of the flow field, an lattice Boltzmann equation (LBE) model developed in Shao et al. (Physical Review E, 89 (2014), 033309) for consideration of density contrast in the momentum equation is used. Moreover, in the present work, the multiple relaxation time collision operator is applied to this LBE to enable simulation of problems with large viscosity contrast or high Reynolds number. For the interface capturing, instead of solving another set of LBE as in many phase field LBMs, the macroscopic Cahn‐Hilliard equation is directly solved by using a weighted essentially non‐oscillatory scheme. In this way, the present hybrid phase field LBM shares full advantages of the phase field LBM while enhancing numerical stability. The ability of the present method to simulate multiphase flow problems with large density contrast is demonstrated by several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytical and numerical studies of the boundary slip in the immersed boundary‐thermal lattice Boltzmann method

We analytically and numerically investigate the boundary slip, including the velocity slip and the temperature jump, in immersed boundary‐thermal lattice Boltzmann methods (IB‐TLBMs) with the two‐relaxation‐time collision operator. We derive the theoretical equation for the relaxation parameters considering the effect of the advection velocity on the temperature jump of the IB‐TLBMs. The analytical and numerical solutions demonstrate that the proposed iterative correction methods without the computational cost of the sparse matrix solver reduce the boundary slip and boundary‐value deviation as effectively as the implicit correction method for any relaxation time. Because the commonly used multi‐direct forcing method does not consider the contributions of the body force to the momentum flux, it cannot completely eliminate the boundary slip because of the numerical instability for a long relaxation time. Both types of proposed iterative correction methods are more numerically stable than the implicit correction method. In simulations of flow past a circular cylinder and of natural convection, the present iterative correction methods yield adequate results without the errors of the velocity slip, the temperature jump, and the boundary‐value deviation for any relaxation time parameters and for any number of Lagrangian points per length. The combination of the present methods and the two‐relaxation‐time collision operator is suitable for simulating fluid flow with thermal convection in the multiblock method in which the relaxation time increases in inverse proportion to the grid size.     

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

Three‐dimensional immersed boundary conditions for moving solids in the lattice‐Boltzmann method

This paper establishes the range of validity for a previously published three‐dimensional moving solid boundary condition for the lattice‐Boltzmann method. This method was reasonably formulated from a mass and momentum balance perspective, but was only verified for a small range of (primarily two‐dimensional) problems. One of the advantages of this boundary condition is that it offers resolution at the sub‐grid scale, allowing for accurate and stable calculation of the force and torque for solids which are moving through a lattice, even for small solid sizes relative to the computational grid size. We verify the boundary condition for creeping flows by comparison to analytical solutions that include both the force and the torque on fixed and moving spheres, and then follow this with comparisons to experimental and empirical results for both fixed as well moving spheres in inertial flows. Finally, we compare simulation results to numerical results of other investigators for the settling of an offset sphere and the drafting–kissing–tumbling of two sedimenting spheres. We found that an accurate calculation of the collision‐operator weighting used to obtain sub‐grid‐scale resolution was necessary in order to prevent spikes in the velocities, forces, and moments when solid objects cross‐computational cells. The wide range of comparisons collected and presented in this paper can be used to establish the validity of other numerical models, in addition to the one examined here. Published in 2007 by John Wiley & Sons, Ltd.     

A lattice Boltzmann method for viscous free surface waves in two dimensions

We propose a new method based on the combination of the lattice Boltzmann equation (LBE) and the kinematic boundary condition (KBC) method to simulate viscous free surface wave in two dimensions. In our method, the flow field is modeled by LBE, whereas the free surface is explicitly tracked by the local height function, which is calculated by the KBC method. The free surface boundary condition (FSBC) for LBE is revised from previous researches. Interpolation‐supplemented lattice Boltzmann (ISLB) method is introduced, which enables our approach to be applied on arbitrary, nonuniform mesh grids. Five cases are simulated respectively to validate the LBE–KBC method: the stationary flow and the solitary waves simulated by the revised‐FSBC are more accurate than the one obtained by the former‐FSBC; numerical results of standing waves show that our method is compatible to the existing two‐dimensional finite‐volume scheme; cases of small amplitude Stokes wave and waves traveling over a submerged bar show good agreement on wave celerity, wavelength, wave amplitude and wave period between numerical results and corresponding analytical solutions and/or experiment data.Copyright © 2012 John Wiley & Sons, Ltd.     

A rotating reference frame‐based lattice Boltzmann flux solver for simulation of turbomachinery flows

In this paper, the newly developed lattice Boltzmann flux solver (LBFS) is developed into a version in the rotating frame of reference for simulation of turbomachinery flows. LBFS is a finite volume solver for the solution of macroscopic governing differential equations. Unlike conventional upwind or Godunov‐type flux solvers which are constructed by considering the mathematical properties of Euler equations, it evaluates numerical fluxes at the cell interface by reconstructing local solution of lattice Boltzmann equation (LBE). In other words, the numerical fluxes are physically determined rather than by some mathematical approximation. The LBE is herein expressed in a relative frame of reference in order to correctly recover the macroscopic equations, which is also the basis of LBFS. To solve the LBE, an appropriate lattice Boltzmann model needs to be established in advance. This includes both the determinations of the discrete velocity model and its associated equilibrium distribution functions. Particularly, a simple and effective D1Q4 model is adopted, and the equilibrium distribution functions could be efficiently obtained by using the direct method. The present LBFS is validated by several inviscid and viscous test cases. The numerical results demonstrate that it could be well applied to typical and complex turbomachinery flows with favorable accuracy. It is also shown that LBFS has a delicate dissipation mechanism and is thus free of some artificial fixes, which are often needed in conventional schemes. Copyright © 2016 John Wiley & Sons, Ltd.     

A new modification of the immersed‐boundary method for simulating flows with complex moving boundaries

In this paper, a new immersed‐boundary method for simulating flows over complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. The present method is based on a finite‐difference approach on a staggered mesh together with a fractional‐step method. It must be noted that the immersed boundary is generally not coincident with the position of the solution variables on the grid, therefore, an appropriate strategy is needed to construct a relationship between the curved boundary and the grid points nearby. Furthermore, a momentum forcing is added on the body boundaries and also inside the body to satisfy the no‐slip boundary condition. The immersed boundary is represented by a series of interfacial markers, and the markers are also used as Lagrangian forcing points. A linear interpolation is then used to scale the Lagrangian forcing from the interfacial markers to the corresponding grid points nearby. This treatment of the immersed‐boundary is used to simulate several problems, which have been validated with previous experimental results in the open literature, verifying the accuracy of the present method. Copyright © 2006 John Wiley & Sons, Ltd.